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% Created on 2008-03-21 by ZHENG Zhong
% Last changed at $Date: 2008-04-28 10:15:35 +0000 (Mon, 28 Apr 2008) $ by $Author: heavyzheng $, $Revision: 39 $
% $HeadURL: http://buggarden.googlecode.com/svn/quant/study_notes/black_scholes_model.tex $
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\chapter{The Black-Scholes Model}

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\section{Lognormal Property of Stock Prices}

The model of stock price behavior assumes that percentage changes in the stock price in a short
period of time are normally distributed. Define:

\begin{my_itemize}
  \item $\mu$: Expected rate of return on stock.
  \item $\sigma$: Volatility of the stock price.
\end{my_itemize}

The mean of the percentage change in time $\Delta t$ is $\mu \Delta t$, and the standard deviation
of this percentage change is $\sigma \sqrt{\Delta t}$. So that:

\begin{equation}
  \frac{\Delta S}{S} \sim \phi(\mu \Delta t, \sigma \sqrt{\Delta t})
  \label{distr_dS_div_S}
\end{equation}

where $\Delta S$ is the change in the stock price $S$ in time $\Delta t$, and $\phi(m, s)$ denotes a
normal distribution with mean $m$ and standard deviation $s$.

@todo The model implies that:

\[ lnS_T - lnS_0 \sim \phi \Big( (\mu - \frac{\sigma^2}{2}) T, \sigma \sqrt{T} \Big) \]

From this it follows that:

\begin{equation}
  ln \frac{S_T}{S_0} \sim \phi \Big( (\mu - \frac{\sigma^2}{2}) T, \sigma \sqrt{T} \Big)
  \label{distr_ln_ST_div_S0}
\end{equation}

and:

\begin{equation}
  lnS_T \sim \phi \Big( lnS_0 + (\mu - \frac{\sigma^2}{2}) T, \sigma \sqrt{T} \Big)
  \label{distr_ln_ST}
\end{equation}

where $S_T$ is the stock price at a future time $T$, and $S_0$ is the stock price at time zero. The
equation above shows that $lnS_T$ is normally distributed. This means that $S_T$ has a lognormal
distribution.

From the properties of the lognormal distribution, the expected value, $E(S_T)$, of $S_T$ is:

\begin{equation}
  E(S_T) = S_0 e^{\mu T}
  \label{E_ST}
\end{equation}

This fits in with the definition of $\mu$ as the expected rate of return. The variance, $var(S_T)$,
is:

\begin{equation}
  var(S_T) = S_0^2 e^{2 \mu T} (e^{\sigma^2 T} - 1)
\end{equation}

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\section{The Distribution of the Rate of Return}

The lognormal property of stock prices can be used to provide information on the probability
distribution of the continuously compounded rate of return earned on a stock between times zero and
$T$. Define the continuously compounded rate of return per annum realized between times zero and $T$
as $\eta$. It follows that:

\[ S_T = S_0 e^{\eta T} \]

so that:

\begin{equation}
  \eta = \frac{1}{T} ln \frac{S_T}{S_0}
\end{equation}

It follows from equation \eqref{distr_ln_ST_div_S0} that:

\begin{equation}
  \eta \sim \phi (\mu - \frac{\sigma^2}{2}, \frac{\sigma}{\sqrt{T}})
  \label{distr_eta}
\end{equation}

Thus, the continuously compounded rate of return per annum is normally distributed with mean
$(\mu - \sigma^2 / 2)$ and standard deviation $(\sigma / \sqrt{T})$. Note that, as $T$ increases,
the standard deviation of $\eta$ declines. This is because we are more certain about the average
return per year over a longer period of time (e.g. 20 years) than we are about the return over a
shorter period of time (e.g. one year).

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\section{The Expected Rate of Return}

Equation \eqref{distr_dS_div_S} shows that $\mu \Delta t$ is the expected percentage change in the
stock price in a very short period of time $\Delta t$. This means that $\mu$ is the expected rate of
return in a very short period of time $\Delta t$. It is natural to assume that $\mu$ is also the
expected continuously compounded rate of return on the stock over a relatively long period of time.
However, this is not the case. The continuously compounded rate of return $\eta$ realized over $T$
is:

\[ \eta = \frac{1}{T} ln \frac{S_T}{S_0} \]

and equation \eqref{distr_eta} shows that the expected value of $\eta$ is $\mu - \sigma^2 / 2$.

\textcolor{red}{@todo: see p237, explain why the expected value of $\eta$ is NOT $\mu$.}

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\section{Volatility}

The volatility of a stock, $\sigma$, is a measure of our uncertainty about the returns provided by
the stock. Stocks typically have a volatility between 20\% and 50\%.

From equation \eqref{distr_eta}, the volatility of a stock price can be defined as the standard
deviation of the return provided by the stock in one year (by setting $T = 1$) when the return is
expressed using continuous compounding.

When $T$ is small, equation \eqref{distr_dS_div_S} shows that $\sigma \sqrt{T}$ is approximately
equal to the standard deviation of the percentage change in the stock price in time $T$. Thus, our
uncertainty about a future stock price increases -- at least approximately -- with the square root
of how far ahead we are looking. For example, the standard deviation of the stock price in four
weeks is approximately twice the standard deviation in one week.

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\section{Concepts Underlying the Black-Scholes-Merton Differential Equation}

The Black-Scholes-Merton differential equation is an equation that must be satisfied by the price of
any derivative dependent on a non-dividend-paying stock. The arguments involve setting up a riskless
portfolio consisting of a position in the derivative and a position in the stock. In absence of
arbitrage opportunities, the return from the portfolio must be the risk-free interest rate $r$.

The reason a riskless portfolio can be set up is that the stock price and the derivative price are
both affected by the same underlying source of uncertainty: stock price movements. In any short
period of time, the price of the derivative is perfectly correlated with the price of the underlying
stock. When an appropriate portfolio of the stock and the derivative is established, the gain or
loss from the stock position always offsets the gain or loss from the derivative position, so that
the overall value of the portfolio at the end of the short period of time is known with certainty.

The assumptions we use to derive the Black-Scholes-Merton differential equation are as follows:

\begin{my_itemize}
  \item The stock price follows the \textcolor{red}{what} process with $\mu$ and $\sigma$ constant.
  \item The short selling of securities with full use of proceeds is permitted.
  \item There are no transactions costs or taxes. All securities are perfectly divisible.
  \item There are no dividends during the life of the derivative.
  \item There are no riskless arbitrage opportunities.
  \item Security trading is continuous.
  \item The risk-free rate of interest $r$ is constant and the same for all maturities.
\end{my_itemize}

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\section{Derivation of the Black-Scholes-Merton Differential Equation}

The stock price process we are assuming is: \textcolor{red}{refer to the section before.}

\begin{equation}
  dS = \mu S dt + \sigma S dz
  \label{dS_continuous}
\end{equation}

Suppose that $f$ is the price of a call option or other derivative contingent on $S$. The variable
$f$ must be some function of $S$ and $t$. Hence, from equation \textcolor{red}{what}:

\begin{equation}
  df = \Big( \frac{\partial f}{\partial S} \mu S
           + \frac{\partial f}{\partial t}
           + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \Big) dt
     + \frac{\partial f}{\partial S} \sigma S dz
  \label{df_continous}
\end{equation}

The discrete versions of the above equations are:

\begin{equation}
  \Delta S = \mu S \Delta t + \sigma S \Delta z
  \label{dS_discrete}
\end{equation}

and

\begin{equation}
  \Delta f = \Big( \frac{\partial f}{\partial S} \mu S
                 + \frac{\partial f}{\partial t}
                 + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2
             \Big) \Delta t
           + \frac{\partial f}{\partial S} \sigma S \Delta z
  \label{df_discrete}
\end{equation}

where $\Delta S$ and $\Delta f$ are the changes in $f$ and $S$ in a small time interval $\Delta t$.
From It\^o's lemma, the Wiener processes underlying $f$ and $S$ are the same. In other words, the
$\Delta z$ ($= \epsilon \sqrt{\Delta t}$) in equations \eqref{dS_discrete} and \eqref{df_discrete}
are the same. It follows that, by choosing a portfolio of the stock and the derivative, the Wiener
process can be eliminated.

The appropriate portfolio is as follows:

\begin{my_itemize}
  \item $-1$: derivative.
  \item $+(\partial f / \partial S)$: shares of stock.
\end{my_itemize}

The holder of this portfolio is short one derivative and long an amount $\partial f / \partial S$ of
shares. Define $\Pi$ as the value of the portfolio. By definition,

\begin{equation}
  \Pi = -f + \frac{\partial f}{\partial S} S
  \label{pi}
\end{equation}

The change $\Delta \Pi$ in the value of the portfolio in the time interval $\Delta t$ is given by:

\begin{equation}
  \Delta \Pi = -\Delta f + \frac{\partial f}{\partial S} \Delta S
  \label{dpi_discrete}
\end{equation}

Substituting equations \eqref{dS_discrete} and \eqref{df_discrete} into equation
\eqref{dpi_discrete} yields:

\begin{equation}
  \Delta \Pi = \Big( - \frac{\partial f}{\partial t}
                     - \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2
               \Big) \Delta t
  \label{dpi_no_dz_discrete}
\end{equation}

Because this equation does not involve $\Delta z$, the portfolio must be riskless during time
$\Delta t$. The assumptions listed in the preceding section imply that the portfolio must
instantaneously earn the same rate of return as other short-term risk-free securities. Otherwise,
there will be an arbitrage opportunity. It follows that:

\[ \Delta \Pi = r \Pi \Delta t \]

where $r$ is the risk-free interest rate. Substituting from equations \eqref{pi} and
\eqref{dpi_no_dz_discrete}, we obtain:

\[
  \Big( - \frac{\partial f}{\partial t}
        - \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2
  \Big) \Delta t
  =
  r \Big( -f + \frac{\partial f}{\partial S} S \Big) \Delta t
\]

so that:

\begin{equation}
    \frac{\partial f}{\partial t}
  + r S \frac{\partial f}{\partial S}
  + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2}
  = r f
  \label{bsm}
\end{equation}

Equation \eqref{bsm} is the Black-Scholes-Merton differential equation. It has many solutions,
corresponding to all the different derivatives that can be defined with $S$ as the underlying
variable. The particular derivative that is obtained when the equation is solved depends on the
boundary conditions that are used. These specify the values of the derivative at the boundaries of
possible values of $S$ and $t$. In the case of a European call option, the key boundary condition
is:

\[ f = max(S - K, 0) ~ when ~ t = T \]

In the case of a European put option, it is:

\[ f = max(K - S, 0) ~ when ~ t = T \]

Note that the portfolio used in the derivation of equation \eqref{bsm} is not permanently riskless.
It is riskless only for an infinitesmally short period of time. As $S$ and $t$ change,
$\partial f / \partial S$ also changes. To keep the portfolio riskless, it is therefore necessary to
frequently change the relative proportions of the derivative and the stock in the portfolio.

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\section{Black-Scholes Pricing Formulas}

The Black-Scholes formulas for the prices at time zero of a European call option on a non-dividend-
paying stock and a European put option on a non-dividend-paying stock are:

\begin{equation}
  c = S_0 N(d_1) - K e^{-rT} N(d_2)
  \label{bs_call}
\end{equation}

and

\begin{equation}
  p = K e^{-rT} N(-d_2) - S_0 N(-d_1)
  \label{bs_put}
\end{equation}

where:

\begin{equation}
  d_1 = \frac{ ln(S_0/K) + (r + \sigma^2 / 2) T }{ \sigma \sqrt{T} }
\end{equation}

\begin{equation}
  d_2 = \frac{ ln(S_0/K) + (r - \sigma^2 / 2) T }{ \sigma \sqrt{T} } = d_1 - \sigma \sqrt{T}
\end{equation}

The function $N(x)$ is the cumulative probability distribution function for a standardized normal
distribution. In other words, it is the probability that a variable with a standard normal
distribution, $\phi(0, 1)$ will be less than $x$. The variables $c$ and $p$ are the European call
and European put price, $S_0$ is the stock price at time zero, $K$ is the strike price, $r$ is the
continuous compounded risk-free rate, $\sigma$ is the stock price volatility, and $T$ is the time to
maturity of the option.

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\section{Derivation of the Black-Scholes Formulas Using Risk-Neutral Valuation}

One way of deriving the Black-Scholes formulas is by solving the differential equation \eqref{bsm}
subject to the boundary conditions of a European call or put option. Another approach is to use
risk-neutral valuation. Consider a European call option. The expected value of the option at
maturity in a risk-neutral world is:

\[ \hat E \Big( max(S_T - K, 0) \Big) \]

where $\hat E$ denotes the expected value in a risk-neutral world. From the risk-neutral valuation
argument, the European call option price, $c$, is this expected value discounted at the risk-free
rate of interest, that is:

\begin{equation}
  c = e^{-rT} \hat E \Big( max(S_T - K, 0) \Big)
  \label{call_risk_neutral}
\end{equation}

We will introduct a lemma which will be used to derive the Black-Scholes formulas. It can be proved
that, if $V$ is lognormally distributed and the standard deviation of $ln V$ is $s$, then:

\begin{equation}
  E \Big( max(V - K, 0) \Big) = E(V) N(d_1) - K N(d_2)
  \label{lemma_for_bs}
\end{equation}

where:

\[ d_1 = \frac{ ln \Big( E(V)/K \Big) + s^2/2 }{s} \]
\[ d_2 = \frac{ ln \Big( E(V)/K \Big) - s^2/2 }{s} \]

and $E$ denotes the expected value.

In equation \eqref{call_risk_neutral}, under the stochastic process assumed by Black-Scholes, $S_T$
is lognormal. Also from equations \eqref{distr_ln_ST} and \eqref{E_ST}, $\hat E(S_T) = S_0 e^{rT}$
and the standard deviation of $ln S_T$ is $\sigma \sqrt{T}$. The lemma \eqref{lemma_for_bs} implies
that:

\[ c = e^{-rT} \Big( S_0 e^{rT} N(d_1) - K N(d_2) \Big) \]

or:

\[ c = S_0 N(d_1) - K e^{-rT} N(d_2) \]

where:

\[ d_1 = \frac{ ln \Big( \hat E(S_T) / K \Big) + \sigma^2 T/2 }{ \sigma \sqrt{T} }
       = \frac{ ln(S_0/K) + (r + \sigma^2 / 2) T }{ \sigma \sqrt{T} } \]

and:

\[ d_2 = \frac{ ln \Big( \hat E(S_T) / K \Big) - \sigma^2 T/2 }{ \sigma \sqrt{T} }
       = \frac{ ln(S_0/K) + (r - \sigma^2 / 2) T }{ \sigma \sqrt{T} } \]


To provide an interpretation of the terms in equation \eqref{bs_call}, we note that it can be
written:

\[ c = e^{-rT} \Big( S_0 N(d_1) e^{rT} - K N(d_2) \Big) \]

The expression $N(d_2)$ is the probability that the option will be exercised in a risk-neutral
world, so that $K N(d_2)$ is the strike price times the probability that the strike price will be
paid. The expression $S_0 N(d_1) e^{rT}$ is the expected value of a variable that equals $S_T$ if
$S_T > K$ and is zero otherwise in a risk-neutral world.

When the Black-Scholes formula is used in practice, the interest rate $r$ is set equal to the zero-
coupon risk-free interest rate for a maturity $T$. This is theoretically correct when $r$ is a known
function of time. It is also theoretically correct when the interest rate is stochastic providing
the stock price at time $T$ is lognormal and the volatility parameter is chosen appropriately.

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\section{Implied Volatilities}

The one parameter in the Black-Scholes pricing formulas that cannot be directly observed is the
volatility of the stock price. In practice, traders usually work with what are known as implied
volatilities. These are the volatilities implied by option prices observed in the market.

It is not possible to invert equation \eqref{bs_call} or \eqref{bs_put} so that $\sigma$ is
expressed as a function of $S_0$, $K$, $r$, $T$, and $c$ or $p$. However, an iterative search
procedure can be used to find the implied $\sigma$.

Implied volatilities are used to monitor the market's opinion about the volatility of a particular
stock. Traders like to calculate implied volatilities from actively traded options on a certain
asset and interpolate between them to calculate the appropriate volatility for pricing a less
actively traded option on the same stock. It is important to note that the prices of deep-in-the-
money and deep-out-of-the-money options are relatively insensitive to volatility. Implied
volatilities calculated from these options tend, therefore, to be unreliable.

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